Alpha Spending On The Event Information Scale

Why Spend Alpha on an Event-Information Scale?

A group sequential logrank design may examine treatment effects after successive numbers of events. Each interim analysis creates an additional opportunity to cross an efficacy boundary, so the nominal threshold at any one look cannot generally be treated as if it were the only test. An alpha-spending rule manages that repeated opportunity by specifying how much of the total type I error may have been spent by each information fraction.

The relevant scale is information, not simply the number of participants enrolled or the calendar time. For a planned maximum information \(I_{\max}\), the information fraction at look \(j\) is \(t_j=I_j/I_{\max}\), where \(I_j\) is the information available at that analysis. In an event-driven logrank design, event counts often help determine \(I_j\), but the information fraction is not automatically identical to the fraction of participants enrolled. The introductory material on information fractions from event information establishes how to obtain the relevant information quantities.

Definition: An alpha-spending function \(A(t)\) gives the cumulative probability, under the global null hypothesis, of crossing an efficacy boundary by information fraction \(t\). For a one-sided design with total type I error \(\alpha\), it is nondecreasing, satisfies \(A(0)=0\), and has \(A(1)=\alpha\). At look \(j\), the cumulative target is \(A(t_j)\); the new amount spent since the previous look is \(A(t_j)-A(t_{j-1})\).

These are crossing probabilities under the null, not probabilities that the null hypothesis is true. Also, the cumulative target \(A(t_j)\) is not usually the marginal tail probability \(P(Z_j\ge b_j)\) at look \(j\). Earlier looks and the correlation between sequential statistics affect the probability of crossing by look \(j\), so a later efficacy boundary must be calibrated jointly with the earlier boundaries.

From Spending Targets to Efficacy Boundaries

Write \(Z_j\) for the one-sided, standardized logrank statistic at look \(j\), with larger values favoring treatment benefit under the chosen direction of the test. Let \(b_j\) be its upper efficacy boundary. The design rejects \(H_0\) for efficacy if any observed statistic crosses its boundary: \(Z_j\ge b_j\) at one or more looks. Under the canonical joint distribution, the statistics have standard normal marginal distributions and, for \(i<j\), correlation \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\). This relationship requires the usual information-based normal approximation and an appropriate sequential logrank construction; independent-increment conditions and the analysis rules must be credible for the planned monitoring.

The boundary at the first look is simple: \(P(Z_1\ge b_1)=A(t_1)\), so \(b_1=\Phi^{-1}(1-A(t_1))\). At a later look, the defining equation is

P_0\left(\bigcup_{k=1}^{j}\{Z_k\ge b_k\}\right)=A(t_j).

Here \(P_0\) denotes probability under \(H_0\). Holding the already calibrated boundaries \(b_1,\ldots,b_{j-1}\) fixed, choose \(b_j\) so the equation holds. The marginal nominal level at look \(j\) is then \(1-\Phi(b_j)\). The increment in cumulative error spent at that look is \(A(t_j)-A(t_{j-1})\), which is generally different from this marginal nominal level.

The reason is that a crossing at the current look counts only if no earlier boundary was crossed. In a two-look design, for example, the probability of crossing by look 2 is

P_0(Z_1\ge b_1)+P_0(Z_1<b_1,\ Z_2\ge b_2).

This decomposition both justifies the calibration rule and gives a practical way to calculate a later boundary. For more looks, multivariate normal integration or a validated group sequential calculation is typically used.

A Common Spending Choice

A one-sided Lan-DeMets O’Brien-Fleming-type spending function is

A(t)=1-\Phi\left(\frac{z_{1-\alpha}}{\sqrt{t}}\right),\qquad 0<t\le 1,

where \(z_{1-\alpha}=\Phi^{-1}(1-\alpha)\). It spends little alpha early and approaches \(\alpha\) at \(t=1\). This function specifies cumulative spending, not every look’s boundary by itself: after the first look, the boundary must still be calibrated using the joint distribution. Other spending shapes are possible. A more even spending schedule tends to allow less stringent early boundaries, while a schedule that spends very little early imposes more stringent early boundaries. The choice is a design decision made before examining interim treatment results.

Worked Example: Spending at an Early Event-Information Look

Suppose a hypothetical one-sided logrank design has total \(\alpha=0.025\), and its first analysis occurs at event information fraction \(t_1=0.50\). Use the O’Brien-Fleming-type spending function to find cumulative alpha spent and the first efficacy boundary.

First, \(z_{1-\alpha}=z_{0.975}=1.960\). The cumulative spending target is

A(0.50)=1-\Phi\left(\frac{1.960}{\sqrt{0.50}}\right) =1-\Phi(2.772) \approx 0.002787.

There are no earlier looks, so this is also the first-look marginal tail probability. The boundary is \(b_1=2.772\), since \(P(Z_1\ge 2.772)\approx0.002786\). Thus the hypothetical study may stop for efficacy at this look if its observed one-sided logrank statistic is at least 2.772. The stringent threshold reflects that only about 0.002787 of the total 0.025 type I error has been spent by halfway through the information.

Using Realized Event Information

Event-driven monitoring does not guarantee that the information fraction at a scheduled analysis equals its planned value. Censoring, the observed event pattern, and risk-set composition can change the information available at a cutoff. If the design specifies spending as a function of information, the analysis uses the realized \(t_j=I_j/I_{\max}\) under the prespecified information calculation, rather than silently substituting the planned fraction.

Worked Example: Updating Cumulative Spending at Two Looks

Consider a hypothetical design with a maximum of 300 planned events, total one-sided \(\alpha=0.025\), and O’Brien-Fleming-type spending. Analyses occur after 120 and 225 events. For this illustration, assume the information calculation supports using event fractions as the information fractions, so \(t_1=120/300=0.40\) and \(t_2=225/300=0.75\).

At the first look, \(z_{0.975}=1.960\), giving

A(0.40)=1-\Phi\left(\frac{1.960}{\sqrt{0.40}}\right) =1-\Phi(3.099)\approx0.00097.

Thus \(b_1=3.099\). At the second look,

A(0.75)=1-\Phi\left(\frac{1.960}{\sqrt{0.75}}\right) =1-\Phi(2.263)\approx0.0118.

The additional cumulative error target between the looks is \(0.0118-0.00097=0.0108\), to the displayed precision. But \(2.263\) is not the second efficacy boundary: it is the normal quantile associated with the cumulative spending target viewed as a single-test tail area. The actual \(b_2\) must satisfy

P_0(Z_1\ge3.099,\ Z_2\text{ unrestricted}) + P_0(Z_1<3.099,\ Z_2\ge b_2) =0.0118,

with \(\operatorname{Corr}(Z_1,Z_2)=\sqrt{0.40/0.75}\approx0.730\). A numerical joint-normal calculation gives \(b_2\); the nominal marginal level at look 2 is then \(1-\Phi(b_2)\). The important arithmetic result here is the spending target and its increment, not an incorrect identification of the second boundary with 2.263.

Calibrating a Later Nominal Level

The following two-look calculation makes the distinction explicit. For two looks with fractions \(t_1=0.50\) and \(t_2=1.00\), the canonical correlation is \(\sqrt{0.50}=0.7071\). Suppose a hypothetical design uses the same upper boundary \(b\) at both looks and chooses it to give cumulative one-sided type I error 0.025 at the final look. The cumulative crossing probability is

1-\Phi(b)+ \int_b^\infty \phi(y)\, \Phi\left(\frac{b-0.7071y}{\sqrt{1-0.7071^2}}\right)\,dy.

The first term is the probability of crossing at look 1. In the integral, \(y\) is the second-look statistic and the conditional normal probability is the chance that \(Z_1<b\) given \(Z_2=y\). Together they count the disjoint possibilities of an early crossing or a first crossing at look 2. Numerical solution sets this expression equal to 0.025 and gives \(b\approx2.175\). The nominal marginal tail probability at either look is \(1-\Phi(2.175)\approx0.0148\), while the cumulative type I error by the final look is 0.025. The difference arises because crossing opportunities are correlated and an early crossing already contributes to the cumulative error.

Worked Example: Interpreting the Calibrated Boundary

For the two-look design above, suppose the hypothetical observed statistics are \(Z_1=2.10\) and \(Z_2=2.18\). The common boundary is approximately 2.175.

At look 1, \(2.10<2.175\), so the efficacy boundary is not crossed. At look 2, \(2.18>2.175\), so the boundary is crossed and the design declares efficacy, assuming it reached that look. This decision is about the prespecified group sequential rejection rule; it is not a conclusion that \(H_0\) is true. If instead the second statistic had been 2.25, it would cross the boundary, provided the earlier analysis had not already stopped the trial.

The nominal tail probability \(0.0148\) describes a single standard normal statistic exceeding 2.175 under its marginal null distribution. It is not the overall type I error for the two-look procedure. That overall error is the joint crossing probability, calibrated to 0.025.

Checks, Alternatives, and Limitations

Several checks help identify a faulty calculation before it affects an interim decision. Confirm that the information fractions are ordered and lie between 0 and 1; that \(A(t_j)\) is nondecreasing; and that the final cumulative target equals the design’s total \(\alpha\). Check that each later boundary was solved using all prior boundaries and the sequential covariance structure. Finally, verify that event and censoring data used to calculate information correspond to the analysis cutoff. Using \(\Phi^{-1}(1-A(t_j))\) as the later boundary, or treating successive logrank statistics as independent, is not a substitute for joint calibration.

Alpha spending is useful when the exact timing of information looks may vary, because the rule can be evaluated at realized information fractions while controlling cumulative null crossing probability under its assumptions. It does not remove the need to prespecify the spending function, the analysis direction, the information calculation, or the stopping rule. If the canonical approximation is poor, information estimates are unreliable, or the actual monitoring departs from the prespecified rule, the nominal calibration may not deliver the intended type I error. Discrete event timing and departures from planned information also warrant explicit evaluation rather than an automatic assumption that the design remains unchanged.

A fixed-look design can instead specify a separate set of boundaries for a fixed schedule, and a simple Bonferroni approach can allocate marginal error probabilities conservatively across looks. Alpha spending differs in that it targets cumulative crossing probabilities under the joint sequential distribution, and it can accommodate analyses at varying information fractions. Neither approach makes the event count itself a universally exact measure of information: the logrank information calculation and its assumptions still matter.

Wrong reasoning: “Cumulative spending by look 2 is 0.0118, so the second boundary is the one-look 0.0118 quantile.”
Correction: That quantile ignores the chance of having crossed at look 1. Calibrate the look-2 boundary so the probability of crossing at either look equals the cumulative target.
Wrong reasoning: “The trial has enrolled 75% of participants, so its information fraction is 0.75.”
Correction: Use the specified event-information calculation. Enrollment fraction and information fraction need not agree.

Key takeaway: The spending function sets cumulative type I error targets on the event-information scale. At the first look, the target gives a marginal boundary directly; at later looks, the nominal efficacy threshold must be calibrated jointly with the earlier logrank statistics.

Check Your Understanding

Answer each question before checking the solutions.

In a one-sided design, what is the distinction between cumulative spending \(A(t_j)\) and the marginal nominal level \(1-\Phi(b_j)\)?
A hypothetical design has \(\alpha=0.025\) and O’Brien-Fleming-type spending at \(t=1\). What cumulative alpha has been spent by then?
Why is the second-look boundary not generally \(\Phi^{-1}(1-A(t_2))\)?
For information fractions \(t_1=0.36\) and \(t_2=0.81\), what is the canonical correlation between the two sequential statistics?

Solutions. (1) \(A(t_j)\) is the null probability of crossing at least one boundary by look \(j\); \(1-\Phi(b_j)\) is the marginal probability that the statistic at look \(j\) alone exceeds its boundary. (2) \(A(1)=\alpha=0.025\), by the definition of the spending function. (3) The cumulative target includes earlier crossings, and the statistics are correlated, so the later boundary must solve a joint crossing-probability equation. (4) \(\sqrt{0.36/0.81}=\sqrt{4/9}=2/3\approx0.667\).